EDIT 2022: Siege was removed from the game and fused with Clash to create the Slash mode. Untested myself but looks like people are unlocking fine, so Slash should work.
Simply play 25 total Siege matches to the end, whether you win or lose.
Forfeiting does not matter, and still counts to your total.
Also, the game mode of the day matches that are stylized like siege, do not count. You must play in the standard game titled 'Siege'
Happy hunting!